Deciding between The Law of Sines and The Law of Cosines.

To completely describe a triangle we should give all three side lengths and all three angles.  If we get the right three pieces of information, we can use either The Law of Sines or The Law of Cosines to find the other three.

Angle-Side-Angle or Side-Angle-Angle (ASA OR SAA):  If we have two angle measurements, the third angle is easily found by subtracting the sum of the two known angles from 180°.  We also have one side length and we know what angle is opposite that length so The Law of Sines is the easiest choice.

Example #1: beta = 30°, gamma = 70°, b = 10"

By subtraction, alpha = (180-30-70)° = 80°, so we can find a and c as follows.

10/sin 30° = a/sin 80°, which becomes sin 80°(10)/sin 30° = 19.69615506...  Let's round this to the nearest thousandth and say a ~= 19.696".


10/sin 30° = c/sin 70°, which becomes sin 70°(10)/sin 30° = 18.79385242...  Let's round this to the nearest thousandth and say c ~= 18.794".

Notice that alpha is the largest angle and a is the longest side.  This is always true that the order of the size of the angles will be the same as the order of the size of the opposite sides.  This helps to check your answers roughly to catch big mistakes like dividing when you should have multiplied or vice versa.
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Example #2: beta = 30°, gamma = 70°, a = 10 m


As before, alpha = 80°, so we can find b and c as follows.


b/sin 30° = 10/sin 80°, which becomes sin 30°(10)/sin 80° =  5.077133059... Let's round this to the nearest thousandth and say b ~= 5.077 m.

10/sin 80° = c/sin 70°, which becomes sin 70°(10)/sin 80° = 9.541888941...  Round this to the nearest thousandth and c ~= 9.542 m.
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Side-Angle-Side (SAS): Find the third side with The Law of Cosines, and then go to the SSS instructions.


Example #3: a = 5', c = 6', beta = 120°

b² =5² + 6² - 2(5)(6)cos120° = 25 + 36 - -30 = 91, so b = sqrt(91) ~= 9.539'
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Side-Side-Side (SSS): rearranging The Law of Cosines, we can get any of the cosines of the angles as combination of a, b and c.

Example #4: a = 5', b = sqrt(91)', c = 6'

We already know that beta is 120° from Example #3, but let's check to see how exact we get it using the Law of Cosines and the inverse cosine function.



cosbeta = (a² + c² - b²)/2ac = (25 + 36 - 91)/2(5)(6) = -30/60 = -1/2

The inverse cosine of -1/2 is exactly 120°.

Now let's find alpha

cosalpha = (b² + c² - a²)/2bc = (91 + 36 - 25)/2(6)sqrt(91) = 102/12sqrt(91) ~= .891042111...

On the calculator, we can [2nd][cos][2nd][(-)] and get 26.9955084.... which is to say alpha ~= 26.996°.

Instead of using The Law of Cosines to find gamma, let's just subtract our two angles from 180° and get gamma = (180-120-26.996)° = 33.004°.

If we use Law of Cosines, the answer is the same to the nearest thousandth of a degree.
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Example #5: a = 5', b = 9.539', c = 6'

We already know that beta is 120° from Example #3, but we are using the approximation of b instead of the exact number.  Let's see if it changes our answer.


cosbeta = (a² + c² - b²)/2ac = (25 + 36 - 9.539²)/2(5)(6) ~= -0.49987535...

The inverse cosine of the answer is 119.9917536...°.  Rounding this to the nearest thousandth of a degree gives us 119.992°, so we are off by a little bit.


Now let's find alpha

cosalpha = (b² + c² - a²)/2bc = (9.539² + 36 - 25)/2(6)(9.539) ~= .891013392...


On the calculator, we can [2nd][cos][2nd][(-)] and get 26.99913319.... which is to say alpha ~= 26.999°.

Instead of using The Law of Cosines to find gamma, subtract the two angles from 180° and get gamma = (180-119.992-26.999)° = 33.009°.

If we use Law of Cosines, the answer is the same to the nearest thousandth of a degree.
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Practice problems. Round inexact answers to a thousandth of a unit.

1) alpha = 75°, beta = 45°, c = 12"

Find a, b and gamma.

2) a = 7 m, b = 8 m, c = 9 m


Find alpha, beta and gamma.


3) alpha = 120°, b = 6', c = 4'

Find a, beta and gamma.

Answers in the comments.